Almost Sure

3 May 11

Continuous Semimartingales

A stochastic process is a semimartingale if and only if it can be decomposed as the sum of a local martingale and an FV process. This is stated by the Bichteler-Dellacherie theorem or, alternatively, is often taken as the definition of a semimartingale. For continuous semimartingales, which are the subject of this post, things simplify considerably. The terms in the decomposition can be taken to be continuous, in which case they are also unique. As usual, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}, all processes are real-valued, and two processes are considered to be the same if they are indistinguishable.

Theorem 1 A continuous stochastic process X is a semimartingale if and only if it decomposes as

\displaystyle  X=M+A (1)

for a continuous local martingale M and continuous FV process A. Furthermore, assuming that {A_0=0}, decomposition (1) is unique.

Proof: As sums of local martingales and FV processes are semimartingales, X is a semimartingale whenever it satisfies the decomposition (1). Furthermore, if {X=M+A=M^\prime+A^\prime} were two such decompositions with {A_0=A^\prime_0=0} then {M-M^\prime=A^\prime-A} is both a local martingale and a continuous FV process. Therefore, {A^\prime-A} is constant, so {A=A^\prime} and {M=M^\prime}.

It just remains to prove the existence of decomposition (1). However, X is continuous and, hence, is locally square integrable. So, Lemmas 4 and 5 of the previous post say that we can decompose {X=M+A} where M is a local martingale, A is an FV process and the quadratic covariation {[M,A]} is a local martingale. As X is continuous we have {\Delta M=-\Delta A} so that, by the properties of covariations,

\displaystyle  -[M,A]_t=-\sum_{s\le t}\Delta M_s\Delta A_s=\sum_{s\le t}(\Delta A_s)^2. (2)

We have shown that {-[M,A]} is a nonnegative local martingale so, in particular, it is a supermartingale. This gives {\mathbb{E}[-[M,A]_t]\le\mathbb{E}[-[M,A]_0]=0}. Then (2) implies that {\Delta A} is zero and, hence, A and {M=X-A} are continuous. \Box

Using decomposition (1), it can be shown that a predictable process {\xi} is X-integrable if and only if it is both M-integrable and A-integrable. Then, the integral with respect to X breaks down into the sum of the integrals with respect to M and A. This greatly simplifies the construction of the stochastic integral for continuous semimartingales. The integral with respect to the continuous FV process A is equivalent to Lebesgue-Stieltjes integration along sample paths, and it is possible to construct the integral with respect to the continuous local martingale M for the full set of M-integrable integrands using the Ito isometry. Many introductions to stochastic calculus focus on integration with respect to continuous semimartingales, which is made much easier because of these results.

Theorem 2 Let {X=M+A} be the decomposition of the continuous semimartingale X into a continuous local martingale M and continuous FV process A. Then, a predictable process {\xi} is X-integrable if and only if

\displaystyle  \int_0^t\xi^2\,d[M]+\int_0^t\vert\xi\vert\,\vert dA\vert < \infty (3)

almost surely, for each time {t\ge0}. In that case, {\xi} is both M-integrable and A-integrable and,

\displaystyle  \int\xi\,dX=\int\xi\,dM+\int\xi\,dA (4)

gives the decomposition of {\int\xi\,dX} into its local martingale and FV terms.

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